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Volume 2, Issue 2
Petrov-Galerkin Method with Local Green's Functions in Singularly Perturbed Convection-Diffusion Problems

O. Axelsson, E. Glushkov & N. Glushkova

Int. J. Numer. Anal. Mod., 2 (2005), pp. 127-146.

Published online: 2005-02

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  • Abstract

Previous theoretical and computational investigations have shown high efficiency of the local Green's function method for the numerical solution of singularly perturbed problems with sharp boundary layers. However, in several space variables those functions, used as projectors in the Petrov-Galerkin scheme, cannot be derived in a closed analytical form. This is an obstacle for the application of the method when applied to multi-dimensional problems. The present work proposes a semi-analytical approach to calculate the local Green's function, which opens a way to effective practical application of the method. Besides very accurate approximation, the matrix stencils obtained with these functions allow the use of fast and stable iterative solution of the large sparse algebraic systems that arise from the grid-discretization. The advantages of the method are illustrated by numerical examples.

  • Keywords

convection-diffusion equation, Petrov-Galerkin discretization, Fourier transform, integral equations, iterative solution.

  • AMS Subject Headings

65F10, 65N22, 65R10, 65R20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-2-127, author = {O. Axelsson, E. Glushkov and N. Glushkova}, title = {Petrov-Galerkin Method with Local Green's Functions in Singularly Perturbed Convection-Diffusion Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2005}, volume = {2}, number = {2}, pages = {127--146}, abstract = {

Previous theoretical and computational investigations have shown high efficiency of the local Green's function method for the numerical solution of singularly perturbed problems with sharp boundary layers. However, in several space variables those functions, used as projectors in the Petrov-Galerkin scheme, cannot be derived in a closed analytical form. This is an obstacle for the application of the method when applied to multi-dimensional problems. The present work proposes a semi-analytical approach to calculate the local Green's function, which opens a way to effective practical application of the method. Besides very accurate approximation, the matrix stencils obtained with these functions allow the use of fast and stable iterative solution of the large sparse algebraic systems that arise from the grid-discretization. The advantages of the method are illustrated by numerical examples.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/925.html} }
TY - JOUR T1 - Petrov-Galerkin Method with Local Green's Functions in Singularly Perturbed Convection-Diffusion Problems AU - O. Axelsson, E. Glushkov & N. Glushkova JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 127 EP - 146 PY - 2005 DA - 2005/02 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/925.html KW - convection-diffusion equation, Petrov-Galerkin discretization, Fourier transform, integral equations, iterative solution. AB -

Previous theoretical and computational investigations have shown high efficiency of the local Green's function method for the numerical solution of singularly perturbed problems with sharp boundary layers. However, in several space variables those functions, used as projectors in the Petrov-Galerkin scheme, cannot be derived in a closed analytical form. This is an obstacle for the application of the method when applied to multi-dimensional problems. The present work proposes a semi-analytical approach to calculate the local Green's function, which opens a way to effective practical application of the method. Besides very accurate approximation, the matrix stencils obtained with these functions allow the use of fast and stable iterative solution of the large sparse algebraic systems that arise from the grid-discretization. The advantages of the method are illustrated by numerical examples.

O. Axelsson, E. Glushkov and N. Glushkova. (2005). Petrov-Galerkin Method with Local Green's Functions in Singularly Perturbed Convection-Diffusion Problems. International Journal of Numerical Analysis and Modeling. 2 (2). 127-146. doi:
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